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Theorem rmo5 2582
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmo5 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))

Proof of Theorem rmo5
StepHypRef Expression
1 df-mo 1952 . 2 (∃*𝑥(𝑥𝐴𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
2 df-rmo 2367 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rex 2365 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2366 . . 3 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
53, 4imbi12i 237 . 2 ((∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) → ∃!𝑥(𝑥𝐴𝜑)))
61, 2, 53bitr4i 210 1 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1426  wcel 1438  ∃!weu 1948  ∃*wmo 1949  wrex 2360  ∃!wreu 2361  ∃*wrmo 2362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-mo 1952  df-rex 2365  df-reu 2366  df-rmo 2367
This theorem is referenced by:  nrexrmo  2583  cbvrmo  2589  bdrmo  11404
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